Standard C.2. Pedagogical Knowledge and Practices for Teaching Mathematics

Candidates seeking to become high school mathematics teachers may come into programs with excitement about doing mathematics, but they may be less aware that many of the students they will teach may not share that excitement. Although their inclinations may be to focus on the students most like themselves, they need experiences that help them recognize that teaching is much more than sharing their mathematical knowledge with those few students who are already engaged in learning mathematics. They need to develop commitment to supporting learning by all students and to develop instructional practices that will help them achieve that commitment.

Well-prepared beginning high school mathematics teachers advocate for and ensure that each and every high school student is given the opportunity to learn mathematics in meaningful ways. In particular, well-prepared beginners give serious attention to students who are living in poverty, Latinx, Black, indigenous, and emergent multilinguals, students who have been historically excluded or marginalized in mathematics. All too often, whether intentional or not, secondary mathematics teachers become gatekeepers to students’ opportunities to engage in coursework that will prepare them for a myriad of job opportunities and life events. On the one hand, effective teachers have high expectations for their students. They pose high-cognitive-demand tasks and maintain high levels of cognitive demand by questioning students in meaningful ways and facilitating discourse.  Such practices support their students’ development of relational understandings of mathematics (Skemp, 1976) and understandings of how one concept relates to different forms of mathematics, to themselves, and to the broader world. On the other hand, if a teacher creates an environment in which students engage mainly in memorizing rules and mimicking what the teacher does, then the students will not develop meaningful mathematical knowledge and skills needed for their future success, not only in continuing education or the workforce but also in understanding how mathematics can contribute solutions to broader issues facing society, such as poverty, cancer, or access to adequate housing. Well-prepared beginners have opportunities to understand that such topics are complex and to find ways to highlight that complexity through rigorous mathematical analysis.

Thus, well-prepared beginning high school teachers need to understand that their beliefs about what learning mathematics entails, along with their beliefs about students’ cultural backgrounds, ability levels, gender identities, and other defining characteristics, affect how they interact with students.  Such interactions largely determine how students will engage in mathematics and how much students will advance. Teachers’ beliefs and school policies related to tracking have caused many students to be placed in courses in which the instruction they receive does not enable them to reason and make sense of mathematics; instead those students become less confident in themselves as learners and doers of mathematics and might not see the point of doing mathematics if it does not relate to their current or future lives. A number of scholars have shown that tracking leads to more unfavorable outcomes than positive ones (Horn, 2006; Oakes, 2008; Stiff & Johnson, 2011). Moreover, tracking or placement of students in particular courses is often based on demographic factors more than on students’ knowledge and abilities (Oakes, 2008; Stiff & Johnson, 2011). When teachers of mathematics hold the powerful belief that all students, regardless of race, ethnicity, gender, socioeconomic status, language use, gender performance, immigrant status, or past performance in coursework, can and will succeed in learning important mathematics, they give their students advantages in learning mathematics that will prepare them for their futures.

Vignette 7.2 could be used with teacher candidates to discuss issues related to opportunities to learn. Mathematics teacher educators might ask teacher candidates the following questions to orchestrate a discussion focused on the vignette:

  1. What might be some reasons for the teacher’s taking a hands-off approach to some of the students on this particular day?
  2. Why might some classrooms at the school have such small numbers of students?
  3. What would have been a better approach if the teacher were trying to help the students to develop more autonomy?
  4. What are some strategies that could have been used to ensure that each student was moving forward in his or her learning?

Vignette 7.2. Providing Opportunities to Learn

This excerpt describes events that took place in a high school pre-algebra classroom.

There were four students in the class today: Antonio, Vanessa, Alicia, and David. Three of the students are Black (Antonio, Vanessa, and Alicia) and one is White (David). When the class started, Vanessa asked me for help with her assignment. I assumed that the class would go as it had yesterday, with me working with Vanessa (because she was ahead of the others in the book) and Ms. Smith, the regular classroom teacher, working with the rest of the class. However, this is not what happened. I worked with Vanessa, but Ms. Smith did not work with the other students. David asked for and was given permission to go to the nurse's office. He was gone for most of the period. Antonio appeared to try a few times to get started on his assignment, but he seemed unsure how to begin. He asked Vanessa at least twice to come over to his desk and help him. But Ms. Smith refused to allow Vanessa to work with him. Antonio did not ask Ms. Smith for help, nor did she offer him any assistance. He appeared to get little, if any, work done during this class period. Similarly, Alicia appeared to be trying to answer the questions in the book, but, given what we have seen in the past, I have to wonder how much she understood of what she was doing. Like Antonio, she did not ask Ms. Smith for help, and no assistance was offered. Ms. Smith sat at her desk. She did not speak to Antonio or Alicia except to tell Antonio that he could not get help from Vanessa. At some point Ms. Smith began to make the model staircase that the students would be asked to construct in the next section of the book. When David returned to the classroom, he asked if he could also build a staircase. He worked on the model for the rest of the period. Thus, the class period ended with essentially no instructional interaction between teacher and students. After class, Ms. Smith indicated that she was intentionally taking a hands-off approach in which she would not give help until the students took the initiative to ask for it. This action (or, more accurately, inaction) appeared to be targeted at Antonio. Ms. Smith stated that he makes no effort and only disrupts the class. This appeared to be the justification for her hands-off approach.

We share this example because it was, in fact, representative of a larger pattern. What happened in this class period was not unique in our observations of the teachers' classes. The particular conditions of this class, specifically the small class size, highlight a phenomenon that could be seen elsewhere. We characterize this pattern of action and inaction as "allowing students to fail." As a result of Ms. Smith's hands-off approach, Antonio and Alicia, the only two students in the class in need of help, received no instruction from the teacher for an entire 50-minute class. Yet, this was but a starker example of events that took place in other teachers' classrooms as well. In other cases, students sat for the entire class period, essentially ignored by their teachers. These students were allowed to not work as long as they did not disrupt the class. Students who did not "take responsibility for their own learning" were subject to this hands-off approach.

Note. Adapted from "No Time Like the Present: Reflecting on Equity in School Mathematics" by C. Rousseau and W. F. Tate, 2003, Theory into Practice, 42(3), 210–216. Copyright 2003 by Taylor and Francis, Ltd. Adapted with permission.

Vignette 7.2 may cause some teacher candidates to gasp in disbelief and others to say that they have witnessed such situations in school placements or as students in high school. The discussion of the vignette or similar experiences will cause them to think about students’ opportunities to learn and the role of teacher as an advocate for students, not simply a disseminator of information. Teachers need to engineer classrooms to provide students appropriate content exposure, content coverage, content emphasis, and quality instructional delivery (Tate, 2005). More than just planning for or implementing practices that support students to learn, mathematics teacher candidates need to look for opportunities to measure whether students would agree that they, indeed, have opportunities to learn.